A mixed graph is a graph in which directed and undirected edges between two distinct vertices may exist at most once. To distinguish, we refer an arc to as a directed edge in a mixed graph. An undirected graph is a mixed graph without arcs. For a mixed graph \(G\)  of order \(n\) , let the vertex set of \(G\)  be \(V(G)=[n]=\lbrace 1,2,\ldots ,n\rbrace \) , and the set \(E(G)\)  collects all arcs \(\overrightarrow{ij}\)  and all undirected edges \(ij\)  in \(G\) , where \(i, j\in V(G)\)  are distinct. The size of a mixed graph \(G\)  is defined to be the number of arcs plus twice the number of undirected edges in \(G\) . The adjacency matrix \(A=(a_{ij})\)  of \(G\)  is a square 01-matrix of order \(n\)  defined by \(a_{ij}=1\)  if and only if \({ij}\)  or \(ij\)  in \(E(G)\) . The diagonal matrix \(D^+={\rm diag}(d^+_1, d^+_2, \ldots , d^+_n)\)  is called the out-degree matrix of \(T\) , where \(d^+_i=|\lbrace j~:~{ij}\in E(G){~\rm or~}ij\in E(G)\rbrace |\) . For real number \(\alpha \in [0,1]\) , the matrix \(A_\alpha (G)\)  of \(G\)  is defined to be \(\alpha D^++(1-\alpha )A\) . The concepts of \(A_\alpha \)  matrix of graphs were first introduced by Nikiforov[1]} in 2017 and then liu et al.[2]} start to consider the \(A_\alpha \)  matrix for digraphs. Since \(A_\alpha \)  is nonnegative and it is well known that a nonnegative matrix has a real eigenvalue, let \(\rho _\alpha (G)\)  denote
the largest real eigenvalue \(\rho (A_\alpha (G))\)  of the \(A_\alpha \)  matrix \(A_\alpha (G)\)  of \(G\) , and refer \(\rho _\alpha (G)\)  to as the \(A_\alpha \) -spectral radius, or \(\alpha \) -index of \(G\) . For the previous studies on \(A_\alpha \) -spectral radii of undirected graphs and mixed graphs, see [3]}, [4]}, [5]}, [6]}, [7]}, [8]}.
